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Mathematics LibreTexts

7.3: Unusual Number systems

  • Page ID
    7327
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    Dualtown Number System

    The peoples of Dualtown use only numbers which are 1 and some of the multiples of 2 (even numbers).   \(E=\{1, 2, 4, 6,8 \cdots\}\). Notice that the set \(E\) is closed under multiplication.

    Let's construct a \( 4 \times 4\) multiplication table \(E\).

    \(\times\) \(1\) \(2\) \(4\) \(6\)
    \(1\) \(1\) \(2\) \(4\) \(6\)
    \(2\) \(2\) \(4\) \(8\) \(12\)
    \(4\) \(4\) \(8\) \(16\) \(24\)
    \(6\) \(6\) \(12\) \(24\) \(36\)

    The smallest ten prime numbers in Dualtown are \(2, 6, 10, 14, 18, 22, 26, 30, 34, 38\). Notice that \(36=(6)(6)=(2)(18)\). Thus \(36\) has two different Dualtown prime factorizations.  Hence the Prime Divisibility Theorem  does not hold for the Dualtown number system.